Asymptotic Analysis of Variational and Hamiltonian PDEs

变分偏微分方程和哈密顿偏微分方程的渐近分析

• 批准号: 0412310
• 负责人: Nicholas Ercolani
• 金额: $ 17.94万
• 依托单位: University of Arizona
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-07-01 至 2007-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0412310&HistoricalAwards=false
• 关键词: Asymptotic; Analysis; Variational; Hamiltonian; PDEs; Asymptotic; Analysis; Variational; Hamiltonian; PDEs

The study of pattern forming systems is a fundamental area of scientific investigation in which the tools of modern mathematical analysis (dynamical systems theory, the theory of partial differential equations, asymptotic analysis and probability theory) can be brought to bear on the modelling of physical systems especially near a critical transition in the behavior of the system. For the projects being studied in this proposal the PI is principally interested in patterns that arise when a continuous translational symmetry is reduced, at a critical threshold, to a discrete periodic symmetry resulting in what is often referred to as a "striped" pattern. Such patterns are for instance generic in Rayleigh-Benard convection (RBC) which is a key mechanism in the formation of many weather patterns. In RBC the striped pattern corresponds to the formation, at a criticaltemperature, of periodic "convection rolls" of a uniform characteristic width. A major goal of this research is to study not just the patterns that form at a critical threshold but to characterize the types of defects that arise in these patterns when one is far from threshold. The patterns studied here are stationary and, consistent with that, one may investigate the formation of defects through the minimization of a free energy which depends on a control parameter with a critical value that corresponds to the critical threshold of the physical system. The free energy studied here is known as the "regularized Cross-Newell energy" and is expressedin terms of a phase that locally defines the striped pattern away from defects. From the mathematical point of view the problem is to study the limiting behavior of minimizers to a variational problem as a critical parameter is approached. The determination of such limits is a delicate matter but the PI is developing geometric and analytical tools that should make this determination tractable.A second part of this proposal also involves the asymptotic study of variational problems. The general context of these problems is the study of the statistics of eigenvalues of N x N Hermitean matrices as N becomes large. These statistics are taken with respect to a class of probabilistic expectations which are invariant under the symmetries that preserve the eigenvalues. This general areaof study is generally referred to as random matrix theorywhich has become prominent because of the insights it provides into a wide range of fields including graph theory, number theory and quantum field theory. Indeed the PIand his collaborators recently extablished the existence of a large N expansion for the random matrix partition function which has played a central role over the last twenty years in discussions within the physics community concerning quantum gravity. The key to this analysis is the study of a variational problem that characterizes the asymptotic expected density of eigenvalues and the asymptotics of an associated Riemann-Hilbert problem. The current proposal will study the fine sturcture of this expansionwhich can provide detailed information about counting functions which enumerate graphs on topologically non-trivial surfaces. These counting functions will be determined as solutions to partial differential equations which are continuum limits of Hamiltonian (in fact completely integrable) sytems of ordinary differential equations which describe how the eigenvalue density changes as the probabilistic matrix expectations are varied. The PI will also explorethe behavior of the random matrix partition function as a critical threshold in the variation of these expectations is approached. Physicists have conjectured that this phase transition can be used to determine a viable candidate for calculating field theoretic expectations in two-dimensional quantum gravity. Both parts of this proposal have potential relevance for interestingapplications. The work on pattern formation will make definite predictions on defect formation that can be tested in the laboratory. Indeed some of the predictions emerging from this work are currently being tested by experimentalists. The PI envisions applications to the stability analysis of patterns and some initial studies of the dynamics of wave patterns. In the future one may expect this work to have relevance for modelling defect structure in animal coat patterns (including fingerprints) and the evolution of plantpatterns.The methods and ideas of random matrix theory are having a broad impact; in number theory, combinatorics, random graph theory (or networks), growth processes and multivariate statistics, to name just a few areas of application. It is important that the underlying methods and results of random matrix theory be established with the highest standards ofrigor so that they can be widelyused in these diverse applications. That is an overarching and fundamental goal of this part of the research proposal.

对模式成型系统的研究是科学研究的一个基本领域，其中现代数学分析的工具（动力学系统理论，部分微分方程，渐方性分析和概率理论）可以带给物理系统的建模，尤其是在系统行为中的关键过渡中。对于在此提案中研究的项目，PI主要对在临界阈值下降低连续的翻译对称性的模式感兴趣，从而导致了离散的周期性对称性，从而导致通常称为“条纹”模式。这样的模式是雷利 - 贝纳德对流（RBC）中的通用模式，这是形成许多天气模式的关键机制。在RBC中，条纹模式对应于临界特性宽度的周期性“对流卷”的形成。这项研究的一个主要目的是不仅研究形成临界阈值的模式，而且要表征当一个模式远离阈值时出现的缺陷类型。这里研究的模式是静止的，与此相一致，可以通过最小化自由能的最小化来研究缺陷的形成，该自由能取决于控制参数，其临界值与物理系统的临界阈值相对应。这里研究的自由能称为“正则化跨纽维尔能量”，是局部定义条纹模式远离缺陷的相的术语。从数学的角度来看，问题是研究最小化问题的限制行为，因为接近关键参数。这种限制的确定是一个微妙的问题，但是PI正在开发几何和分析工具，这些工具应该使该确定可进行。该提案的第二部分还涉及变异问题的渐近研究。这些问题的一般背景是研究N X N冬虫矩阵的特征值的统计数据。这些统计数据是关于在保留特征值的对称性下不变的一类概率期望的。这项一般研究通常被称为随机矩阵理论，由于其对广泛领域的见解，包括图理论，数字理论和量子场理论，因此变得很突出。的确，他的合作者最近扩大了随机矩阵分区功能的大量N扩展，该功能在过去二十年中在物理界的讨论中发挥了核心作用。该分析的关键是对特征特征值的渐近预期密度和相关的Riemann-Hilbert问题的渐近性的研究的研究。当前的提案将研究这种扩展的精细坚固，可以提供有关计数功能的详细信息，这些信息列举了拓扑非平凡的表面图。这些计数函数将确定为部分微分方程的解决方案，这些方程是普通微分方程的哈密顿量（实际上是完全可整合的）系统的连续限制，这些方程描述了特征值密度随着概率矩阵期望的变化而变化。 PI还将探索随机矩阵分区函数的行为，作为这些期望变化的关键阈值。物理学家猜想该相变可用于确定在二维量子重力中计算现场理论期望的可行候选者。 该提案的两个部分与有趣的应用都有潜在的相关性。关于模式形成的工作将对可以在实验室中进行测试的缺陷形成做出明确的预测。实际上，目前，实验者正在测试这项工作的一些预测。 PI设想了对模式的稳定性分析的应用以及波模式动力学的一些初步研究。将来，人们可能会期望这项工作与动物外套模式中的缺陷结构（包括指纹）和植物植物的进化具有相关性。随机矩阵理论的方法和思想具有广泛的影响。在数字理论，组合学，随机图理论（或网络），增长过程和多元统计信息中，仅举几个应用领域。重要的是，必须以最高标准的Rigor标准来建立随机矩阵理论的基本方法和结果，以便它们可以在这些不同的应用中被广泛使用。这是研究建议的这一部分的总体和基本目标。